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Let $n$ be a positive squarefree integer, and let $h_n$ denote the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-n})$. Then, is it true that $h_n$ is odd if and only if $n$ is a prime? If yes, then could you please provide a reference to this statement?

I'm new to class field theory and genus theory so I do not know what's been proven or known about this in the literature.

Thank you.

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The condition shouldn't be "$n$ is prime" but "$n$ is either 1, 2, or a prime congruent to 3 mod 4". For instance $\mathbb{Q}(-5)$ has class number 2.

The more general statement that the 2-torsion subgroup of the class group (i.e. the subgroup of elements of order 1 or 2) has order $2^{d-1}$, where $d$ is the number of prime factors of the discriminant. Here is a student project which gives a very detailed proof of this statement, without using any heavy machinery beyond the definitions.

(See also this question for more discussion and references -- in particular Paul Monsky's answer sketches much slicker but less elementary approach via Hilbert's theorem 90.)

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  • $\begingroup$ Leoffler Is there a known characterization of particular class of $n$ for which if the class number is odd then $n$ is always a prime? Say, if $n\equiv11\pmod{24}$, then, $h_n$ is odd if and only if $n$ is a prime? Thanks! $\endgroup$
    – user492144
    Commented Sep 30, 2022 at 6:56
  • $\begingroup$ That is already answered above. If $n$ is squarefree and $3 \bmod 4$, then the implication "$h_n$ is odd iff $n$ is prime" holds. $\endgroup$
    – David Loeffler
    Commented Sep 30, 2022 at 7:13
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    $\begingroup$ (By the way, where I come from, addressing people by their surname alone – without a title prefix – is considered quite rude. In any case, misspelling someone's name is rude in pretty much all parts of the world.) $\endgroup$
    – David Loeffler
    Commented Sep 30, 2022 at 7:16
  • $\begingroup$ Oh, sorry my bad. Where I come from, people don't exist :) $\endgroup$
    – user492144
    Commented Sep 30, 2022 at 7:21

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